The curves x^2 + y^2 + 4x + 6y = 12 and 5y = 4x + 13 are drawn in the coordinate plane. How many times do they intersect?
x^2 + y^2 + 4x + 6y = 12 and 5y = 4x + 13
Rearrange as the second equation as
y = [ 4x + 13 ] / 5
Sub this into the second equation
x^2 + ( [ 4x + 13]/ 5)^2 + 4x + 6 ( [ 4x + 13' / 5) = 12 simplify
x^2 + [ 16x^2 + 104x + 169 ] / 25 + 4x + [ 24x + 78 ] / 5 = 12 multiply through by 25
25x^2 + 16x^2 + 104x + 169 + 100x + 120x + 390 = 300
41x^2 + 324x + 259 = 0 factor as
(41x + 37) ( x + 7) = 0
Set each factor to 0 and solve for x and we get that
x = -37/41 and x = -7
This indicates that there are two intersection points